Multidimensional subwavelength position sensing using a
semiconductor laser with optical feedback
Seth D. Cohen,1,∗ Andrés Aragoneses,2 Damien Rontani,1 M. C. Torrent,2 Cristina Masoller2 and
Daniel J. Gauthier1
1
2
Duke University, Physics Department, BOX 90305, Durham, NC, 27708, USA
Departament de Fı́sica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Colom 11, 08222 Barcelona, Spain
∗
Corresponding author: sdc18@phy.duke.edu
Compiled September 18, 2013
We demonstrate experimentally how to harness quasi-periodic dynamics in a semiconductor laser with dual
optical feedback for measuring subwavelength changes in each arm of the cavity simultaneously. We exploit
the multi-frequency spectrum of quasi-periodic dynamics and show that independent frequency shifts are
mapped uniquely to two-dimensional displacements of the arms in the external cavity. Considering a laser
diode operating at telecommunication wavelength λ ≈ 1550 nm, we achieve an average nanoscale resolution of
c 2013 Optical Society of America
approximately 9.8 nm (∼ λ/160). OCIS codes: (280.3420) Remote sensing and sensors, Laser sensors; (250.5960) Semiconductor lasers.
PZTy
The extreme sensitivity of semiconductor lasers with optical feedback has been studied for its applications to
sensors [1]. For example, the subwavelength sensitivity
to changes in the external cavity length of the feedback has triggered the development of laser-feedback interferometry, also referred as self-mixing interferometry
[2–4], where displacements and vibrations are measured
with subwavelength resolution by tracking changes in
the laser intensity. Other approaches have used changes
in the relaxation-oscillation frequency [5]. These laserbased approaches usually lead to the extraction of a single observable, thus restricting sensing or imaging to one
dimension (1D). As a consequence, to image or sense
complex objects in 2D or 3D, such laser-based systems
require scanning and sequential 1D acquisitions.
Recently, a subwavelength 2D position sensor was realized using a high-speed nonlinear feedback system comprising a radio-frequency (λ ∼ 10−2 m) wave-chaotic
cavity [6]. It was demonstrated that the 2D position of
an object inside the cavity could be determined with subwavelength precision using a linear map that uniquely associates frequency shifts in the quasi-periodic spectrum
of the dynamics with the object position [6]; this method
does not require sequential scanning to deduce 2D information. An open question is whether these frequency
shifts can also occur in the optical domain, thus allowing for the development of novel laser feedback-based devices with the capability for subwavelength, nanoscale,
multidimensional position sensing.
In this Letter, we demonstrate experimentally, using
a semiconductor laser with quasi-periodic dynamics induced by external optical feedback from a cavity with
two arms (referred to as T-cavity), the existence of frequency shifts in the spectrum, similar to those found in
Ref. [6]. We unveil a quadratic mapping between these
frequency shifts with independent, subwavelength translations of each arm of the T-cavity. We show that our
My
Δy
τy
PC
LD
OC
OA
BS
Δx
τx
v(t)
90/10
OI PD
PZTx
Mx
OSC
Fig. 1. (Color online) Two-delay all-optical feedback system for multi-dimensional subwavelength sensing. The
output of a laser diode (LD, Sumitomo SLT4416-DP)
passes through a polarization control (PC), a 90/10 optical coupler (OC), an optical attenuator (OA), and separates using a 50/50 beam splitter (BS). The feedback
strength is set so that it reduces the laser threshold by
∼ 1%. The optical field propagates along two separate
paths with time delays τx,y . Piezoelectric transducers
(PZTx,y , Burleigh PZO-015) translate mirrors Mx,y by
∆x,y and change feedback delay τx,y . After an optical isolator (OI), detection is performed using a 12 GHz photodetector (PD, New Focus 1544-B) and a 40 GS/s, 8 GHz
analog bandwidth, high-speed digital oscilloscope (OSC,
Agilent DSO90804A).
approach leads to a highly sensitive detection scheme
for 2D nanoscale translations.
Our experimental setup is shown in Fig. 1. A singlemode semiconductor laser operating at λ = 1550.8 nm
is subjected to a dual optical feedback with two arms
of lengths x and y, respectively. The propagation distances along each path result in two different time delays,
τx ∼ 55.5 ns and τy ∼ 55.6 ns, with feedback strengths
ηx,y approximately equal and controlled simultaneously
1
v(t) (mV)
5
-40
(a)
(b)
-60 f1
dBm
by a single variable optical attenuator placed before the
beam splitter (see Fig. 1). A polarization controller is
used to ensure coherent feedback to the laser (verified
using L-I curves [7]). Our setup is reminiscent of previous experiments with T-cavity optical feedback [8–10].
The relative values of the time delays are controlled
using two piezoelectric transducers (PZTx,y ) that move
the positions of the mirrors on a nanoscale by ∆x x
and ∆y y, respectively. To estimate the transducers
displacements as a function of the applied voltages, we
calibrate them via interferometry. Based on our calibration measurements, the translations per applied voltages
to PZTx,y are 29.2 ± 0.1 nm/V and 45.8 ± 0.2 nm/V, respectively. We note that this calibration does not include
either the hysteresis or nonlinear response of the PZTs
under applied voltages. To reduce the effects of the PZT
nonlinearity and hysteresis, we bias them at 50 V (out
of a total range of 0 − 150 V) and scan them by at most
3 V about this value. Based on the manufacturers specification, we estimate that the deviation from linearity
is at most 0.3% for our experiment and hence we ignore
these behaviors in the analysis described below.
We carefully adjust the feedback strength such that
the optical intensity displays quasi-periodic oscillations
in time. This occurs when the feedback strength is about
∼ 2%, consistent with previous reports of feedbackinduced quasi-periodicity [11, 12]. A typical optical intensity time series is displayed in Fig. 2a. It shows a fast
oscillatory signal with slow modulation of its amplitude.
The frequency spectrum of the quasi-periodic signal is
given in Fig. 2b. Within the 8 GHz bandwidth of our oscilloscope, it reveals multiple clusters of frequency peaks
ranging from ∼ 1 GHz to ∼ 7 GHz (additional clusters
above 8 GHz may be present but are not visible without
suitable measurement equipments). The fourth cluster
concentrates the largest amount of spectral power because it lies in the vicinity of the relaxation oscillation
frequency measured to be ∼ 6.5 GHz for pumping current I = 23.6 mA. The peak in the ith cluster with the
maximum spectral power is denoted by fi . In Fig. 2c-d,
we show zoomed regions on the spectrum, highlighting
f2 and f4 , respectively. We investigate the relative frequency shifts ∆fi to the values of fi due to subwavelength translations ∆x,y λ of each arm of the dual
optical feedback. These translations induce changes in
the time delays denoted by ∆τx,y = 2∆x,y /c, where c is
the speed of light in free space.
We track the frequencies of the quasi-periodicity as
τx,y are changed simultaneously by translating the positions of mirrors Mx,y . For this, we use computercontrolled, high-voltage power supplies connected to
PZTx,y . The piezoelectric transducers are translated by
steps of ∼ 10 nm within an approximative 100 nm × 100
nm 2D grid. Over this subwavelength region of interest,
the dynamics remains quasi-periodic. Over substantially
larger regions, we observe that the dynamics bifurcates
to a new behavior or undergoes frequency hopping. At
each point of the grid, we measure the frequency shifts
0
f3
f4
f2
-80
-5
0
2
3
t (ns)
4
5
-40
(c)
dBm
-60
1
(d)
2
3 4 5
f (GHz)
6
7
f4
-60
f2
-80
-100
1.9
-100
dBm
-40
1
-80
f (GHz)
2.4
-100
6.25
f (GHz)
6.75
Fig. 2. (Color online) Experimental quasi-periodic dynamics generated by a semiconductor laser with dual optical feedback. (a) Temporal evolution of v(t) generated
by the photo-detector and proportional to the intensity
of the optical field. The pump current is I = 23.6 mA
(the threshold is Ith ∼ 8.0 mA), τx ≈ 55.5, τy ≈ 55.6 ns
(corresponding to ∆x,y = 0), and ∼ 2% of optical intensity fed back in the laser cavity. (b) Power spectral density (PSD) of v(t) in the quasi-periodic regime showing
four frequency clusters, each labelled by their central frequency fi , i = 1, . . . , 4 (f1 = 1.07 GHz, f2 = 2.13 GHz,
f3 = 5.42 GHz, and f4 = 6.49 GHz). (c)-(d) Zooms on
the structures of the second and fourth labeled frequency
clusters later used for subwavelength sensing.
∆fi associated with each frequency fi of the four labeled
clusters.
The nanoscale modification to the cavity induced by
the transducer leads to observed frequency shifts ∆fi on
a kilohertz scale, which is orders of magnitude smaller
compared to frequencies fi . To resolve such fine variations, we use the trigger skew on our digital oscilloscope,
so that the amount of time that the oscilloscope waits between its initial triggering and the acquisition of a waveform is tskew = 5 µs. Using a large trigger skew causes
small frequency shifts ∆fi to be observed as visible phase
shifts ∆ϕi = ∆fi · tskew across the oscilloscope’s delayed
acquisition window.
Using this approach, measurements of ∆fi have a resolution set by the jitter of the waveform. To reduce the
effect of jitter, we acquire (for each point of the 2D grid)
a waveform v̄(t) resulting from the average of 500 time
series. The averaging is realized in real time using the
oscilloscope. We find that averaging over 500 time-series
measurements optimizes the trade-off between noisereduction in the measurement and the inevitable drift
in the apparatus due to temperature fluctuations. Better thermal shielding in future experiments will minimize this non-ideal behavior. To successfully average
2
60
(b)
100
30
50
Δf4 (kHz)
Δf2 (kHz)
(a)
0
tions and levels of curvature, thus suggesting independent (and thus invertible) frequency shifts as a function of the PZT’s translations. To demonstrate independence and inversion, we propose to fit the nonlinear map
(∆f2 , ∆f4 ) → (∆x , ∆y ) by quadratic bivariate functions
X
aij|2,4 ∆ix ∆jy ,
∆f2,4 (∆x , ∆y ) =
(2)
0
-50
-100
-60
60
60
Δ 40
Δ 40
y (n 20
100
y(
m) 0 20 40 60 80
nm 20
60 80 100
0
)
m
) 0 0 20 40 (nm)
(n
Δ
x
Δx
-30
0≤i+j≤2
where the values of the fitted coefficients aij|2,4 obtained
via regression are given in the caption of Fig. 3. The fitted functions of Eq. (2) are numerically invertible so that
predicted values for ∆x,y can be related to a given measure of frequency shifts ∆f2,4 . The root-mean square differences between the predicted and actual subwavelength
translations realized by PZTx,y are 12.3 nm and 6.8 nm,
respectively. This gives an average resolution of 9.8 nm
∼ λ/160. The maximum error between the experimental map and the fit are εx,max = 27.5 nm (∼ λ/60) and
εy,max = 22.7 nm (∼ λ/70), which sets a bound on the
achievable resolution. In our experiments, the frequency
shifts associated with ∆f2,4 yield the best 2D resolution.
A lower bound for the dynamic range for our system,
given by the ratio between the observed range and resolution, is ∼ 10 on the 100 nm×100 nm grid [as we define
it, the dynamic range is ∼ 100 nm (maximum observed
range in each direction) / ∼ 10 nm (resolution)].
We conjecture that room temperature fluctuations
(which are not controlled in our experiment) are the
main source of errors in our measurements. They induce
erratic changes of the path lengths of each arm in the
cavity, thus affecting the frequencies fi with parasitic frequency shifts. Hence, the manifolds displayed in Fig. 3
can be used only once as calibration measurements to
then infer an unknown subwavelength trajectory on the
100 nm × 100 nm grid. Within a ∼5 min time period,
which corresponds to the typical time scale for the drift
in our apparatus, we realize back-to-back acquisitions of
81 calibration points followed by 32 trajectory points.
Using the calibration points, we realize the mapping to
then recover the shape of the trajectory with an average
2D resolution of ∼ λ/160 (for details, see Ref. [13]). Beyond the 5-min time window, temperature fluctuations
prevent us to infer arbitrary sub-wavelength positions
from the calibration.
In summary, we have presented the foundation for realizing a 2D position sensor with nanometer resolution
using a quasi-periodic optical field generated by a semiconductor laser with optical feedback. To the best of
our knowledge, this is the first experimental study of a
T-cavity optical feedback for subwavelength multidimensional position sensing based on frequency shifts in the
quasi-periodic dynamics spectrum. In Ref. [6], Cohen et
al. hypothesize that their technique for subwavelength
position sensing may be scalable to nanoscale using optical feedback. Our work shows that the physical mechanism responsible for subwavelength sensing in the RF
domain, independent quasi-periodic frequency shifts, remains true in the optical domain, thus laying the foun-
Fig. 3. (Color online) Experimental manifold associated
with the quasi-periodic frequency shifts (a) ∆f2 and (b)
∆f4 as a function of ∆x,y . The 2D manifolds are fitted with quadratic functions with coefficients for (a):
a00|2 = −17.3784 ± 1.0834 kHz, a01|2 = −0.0691 ±
0.0358 kHz·nm−1 , a10|2 = 1.2458 ± 0.0562 kHz·nm−1 ,
a11|2 = 0.0058 ± 0.0005 kHz·nm−2 , a02|2 = −0.0041 ±
0.0003 kHz·nm−2 , a20|2 = −0.0095 ± 0.0009 kHz·nm−2
and for (b): a00|4 = −67.7941 ± 3.2683 kHz, a01|4 =
1.2050 ± 0.1079 kHz·nm−1 , a10|4 = 1.8789 ± 0.1694
kHz·nm−1 , a11|4 = 0.0093 ± 0.0014 kHz·nm−2 , a02|4 =
−0.0127 ± 0.0010 kHz·nm−2 , and a20|4 = −0.0140 ±
0.0026 kHz·nm−2 .
over quasi-periodic waveforms, the trigger height vtrig
must also be tuned so that the oscilloscope triggers only
on the maxima of largest amplitudes, corresponding to
regions of the quasi-periodic signal for which the incommensurate frequencies add constructively. Other triggering values, such as vtrig = 0 V, will cause the averaged
waveforms to collapse to zero because of the non-periodic
nature of the quasi-periodic signal. Using a calibrated
quasi-periodic waveform generated by a stable function
generator (Agilent E8267D with 10−3 Hz frequency resolution), our frequency-resolving protocol yields a ±3 kHz
frequency resolution.
We monitor the quasi-periodic frequency shifts ∆fi as
a function of (∆x , ∆y ). We approximate them using a
nonlinear least-square regression to a model of a fourtone quasi-periodic signal of the averaged data from the
delayed acquisition window of the oscilloscope
X
v̄(t) ≈
Ai sin[2π(fi + ∆fi )(t − tskew )], (1)
1≤i≤4
where Ai , ∆fi are the parameters estimated via regression and fi are measured directly with the oscilloscope
to achieve approximately 100 kHz resolution (using timeseries with ∼ 5.25 × 105 points).
As an example, we plot ∆f2 and ∆f4 as a function
of (∆x , ∆y ) over the entire 2D position grid in Figs. 3ab. Interestingly, we do not observe a near-planar evolution of the frequency shifts similar to those observed
in Ref. [6]. Our two-delay optical system produces frequency shifts that are mapped onto a smooth manifold
with non-zero curvature.
The two manifolds exhibit different ranges of varia3
dation for an all-optical implementation of their setup.
In the future, we envision several directions of dynamical phenomenon. Similar to Ref. [6], the delayed feedback
of the optical system can be coupled into a wave-chaotic
optical cavity [14] such that it interacts with a contained
subwavelength scatterer. In its present form, our system
may also be used as a multidimensional sensor similar
to those developed for monitoring of chemical concentrations [15] or nano-particles [16]. Lastly, the Hopf frequency of a semiconductor laser with feedback has been
studied with respect to the external cavity length [11];
developing theories for the quasi-periodic frequencies in a
T-cavity subject to small delay changes may give deeper
insight into our observations.
S.D.C, D.R, and D.J.G gratefully acknowledge the financial support of the U.S. Office of Naval Research
grant # N000014-07-0734. A.A., M.C.T., and C.M.
gratefully acknowledge the financial support of grant #
FA8655-12-1-2140 from EOARD, # FIS2012-37655-C0201 from the Spanish MCI, and # 2009 SGR 1168 from
the Generalitat de Catalunya. C. M. also acknowledges
support from the ICREA Academia program.
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